The first elephant in the United States (since the extinction of the mammoth and mastodon, anyway, and barring the surreptitious smuggling of an elephant to the colonies by some mad noble who told no one of his acquisition) arrived in New York in April 1796.

After its arrival, the young female was taken on tour around the United States, from New York to Boston to Philadelphia. In 1804, the elephant was bought by Hackaliah Bailey (who would later, with P.T. Barnum, found the Barnum and Bailey Circus). In 1816, the elephant, now named Old Bet, was killed in Maine by a local farmer who thought his neighbors shouldn't be spending their money just to see an animal.

To return to your question, Old Bet was definitely in Philadelphia while it served as the national capital. There's no evidence that President Washington saw the animal, but he had to be aware of it - Old Bet drew huge crowds wherever she went. Indeed, she was so intensely popular that "seeing the elephant" became a common phrase for experiencing something extraordinary you couldn't fully explain to someone who wasn't there. (It was used during the Civil War to describe entering combat for the first time.)

We do know that Thomas Jefferson saw Old Bet in Philadelphia in March 1797, so if Washington didn't see the elephant, we don't have to go too far to find a president that did.